The document evaluates three mathematical functions - Logistic, Gompertz, and Richards - for fitting growth data of normal and malformed mango panicles. It finds that the Richards function provides the best fit with the lowest standard error and highest correlation coefficient. The Richards function is more flexible than the other two functions and can successfully model panicle growth under different treatments and conditions. It is superior for summarizing the growth data of both normal and malformed mango panicles.

1. J. Appl. Hort., 2(1):62-64, January-June, 2000
Use of Logistic, Gompertz and Richards functions for fitting normal
and malformed mango panicle growth data
Shailendra Rajan
Central Institute for Subtropical Horticulture, PO Kakori, Rehmankhera, Lucknow-227I07
Logistic, Gompertz and Richards functions were used for examining their suitability in fitting growth data of malformed and normal mango
panicles. The functional analysis of data indicated that Richards function was a suitable model for summarising panicle growth data.
The model was found to be superior to logistic and Gompertz because of its greater flexibility. The study revealed that Richards function
can be successfully used for simulating panicle growth under different treatments and conditions.
Keywords: Logistic, Gompertz, Richards functions, growth, malformed, mango panicles, functional
analysis
Introduction
Many attempts have been made to simulate limited
growth curves by mathematical functions either
accounting for their form through certain growth
process or to obtain any relatively simple equation,
which contain essence of numerical data. Limited
attempts have been made to explain the growth of
mango panicle through equations, which are used for
describing growth. Mango panicles follow asymptotic
growth curve, which can be fitted by using
monomolecular, logistic and Gompertz equations
(Rajan and Majumder, 1995). Sukhvibul et al. (1999)
used the regression model y= a + bj(l+exp[-(x-c)jdJ)
for describing the effect of temperature on panicle
growth. However Richards growth function, which is
being used in biological studies for explaining growth
of the plant parts and whole plants (Causton et al.,
1978), has not been used for modelling growth of
mango plant parts. Usefulness of this function for
modelling growth of different plant species and for
several process and has been established (Pienaar
and Turnbull, 1973; Moore et al., 1988, Lalancette
et al., 1988)
The background of this biological equation has been
described in detail (Richard, 1969, Causton et al.,
1978). The present paper deals with the evaluation of
the sigmoid models including Richards function for
studying growth pattern of malformed and normal
panicles.
Materials and methods
The linear growth of Amrapali mango panicle was
measured in terms of length of the main axis.
Observations were recorded on the tagged panicles at
three days interval, starting from the bud-burst
stage. The main axis was measured from the
juncture of shoot and panicle on the twigs examined
for protu berances and axilary buds for the selection
of normal and malformed panicles (Rajan, 1987).
Finally the panicles were marked as malformed and
normal when they attained about 3 cm growth. Ten
panicles, each in normal and malformed category,
were selected for recording growth data. Mean
growth data of ten panicles was used for fitting the
models. Initial estimates of the parameters were
obtained as suggested by Draper and Smith (1981)
and the estimates of nonlinear parameters were
obtained by applying Marquard algorithm (Marquard,
1963). Cubic spline was used for interpolating the
data for generating mean growth data at 5 days
interval for malformed and normal panicles.
Following equations were used for fitting the models
1= a dl abee-cl
(l+be-
CI
)' dt (I +be-CI
)2 I
and estimating growth rate of panicle
Logistic
Where, l= length of panicle main axis at the time t,
-e(b-ct) dl (b-ct) -(b-ct)
I = ae ,- = eae .e
dt
a is the measure of final size, band c are the
constants
Gompertz
Where l= length of panicle mail).axis at the time t, a
dl
dt
aee(b-cl)
C+"')m(l + e(b-ct)) ---;;;-
is the measure of final size, band c are the constants
Richards
Where, l= length of panicle main axis at the time t,
a is the measure of final size, b, c and m are the
constants. The constant m lie in the range of -1 ~ m
~ 00 but m "* O. dl/ dt is the absolute growth rate.
Results and discussion
All the three sigmoid growth functions, uiz., logistic,
Gompertz and Richards, used for fitting panicle
growth data were found to be appropriate for both

2. Logistic. Gompert: and Richards functions for fitting mango panicle growth data
the cases i.e. normal and malformed panicles.
However, Richards function provided a better fit in
terms of standard error and correlation coefficient
values (Table 1). Richards function was found
superior to explain the data of both types of panicles.
The Richards function yielded the parameters a, b, c,
Table 1. Parameter estimates of growth models for mango panicles
63
and m of which a and m can be considered
biologically meaningful. Parameter a, gives the
asymptotic maximum size of the panicle and m
describes the shape of the growth curve. As the value
of m increases there is an increasing duration of
panicle growth when its growth rate is almost
constant implying approximate exponential growth.
Model a b c m SE r
1.0716 0.9979
1.6194 0.9947
1.7692 0.9936
0.9093 0.9952
1.3548 0.9882
2.0780 0.9721
Normal panicle
Richard 34.0127
36.2971
36.4857
15.1053
79.8551
3.0319
Logistic
Gompertz
Malformed panicle
Richard
Logistic
Gompertz
20.0989
22.9187
31.7133
14.8010
24.1705
1.2940
0.5284
0.1868
0.1476
4.1301
0.4772
0.1280
0.0500
5.1611
Table 2. Estimated values of panicle growth and absolute growth rate estimated by different models
Days Normal panicle Malformed panicle
Panicle growth estimates Growth rate (cm/day) Panicle growth estimates Growth rate(cm/day)
PL L G R L G R PL L G R L G R
0 0.8516 0.4489 0.0000 0.8776 0.0828 0.0000 0.1123 1.5373 0.9105 0.8265 1.1421 0.1119 0.1507 0.1056
5 2.2483 1.1210 0.0018 1.6639 0.2030 0.0026 0.2129 1.8776 1.6677 1.8517 1.8134 0.1980 0.2630 0.1677
to 3.2754 2.7231 0.3187 3.1546 0.4706 0.2230 0.4036 2.1326 2.9695 3.4708 2.8793 0.3309 0.31}39 0.2662
15 4.5000 6.2102 3.7831 5.9801 0.9617 1.2654 0.7646 3.4110 5.0462 5.6615 4.5713 0.5038 0.4877 0.4225
20 10.0296 12.5004 12.3463 11.3112 1.5311 1.9744 1.4319 6.2176 7.9934 8.2879 7.2516 0.6665 0.5560 0.6671
25 23.0000 20.7.646 21.7337 20.7840 1.6600 1.6617 2.3115 13.2200 11.5494 11.1518 11.4031 0.7335 0.5827 0.9979
30 29.9631 28.0519 28.4809 30.9668 1.1905 1.0411 1.2728 16.6369 15.0882 14.0519 16.6744 0.6600 0.5718 0.9539
35 33.2774 32.5390 32.4111 33.7409 0.6294 0.5664 0.1407 18.8750 17.9951 16.8237 19.5645 0.4950 0.5332 0.2349
40 34.1903 34.7214 34.4774 33.9930 0.2816 0.2881 0.0104 20.2561 20.0295 19.3560 20.0458 0.3233 0.4778 0.0251
45 34.3193 35.6612 35.5113 34.0113 0.1167 0.1419 0.0007 20.3387 21.2990 21.5895 20.0939 0.1927 0.4150 0.0023
50 34.8000 36.0445 36.0165 34.0126 0.0469 0.0688 0.0001 20.5000 22.0353 23.5060 20.0984 0.1088 0.3519 0.0002
PL= Panicle length value interpolated cubic spline for 5 days interval, L= Logistic, G= Gompertz, R= Richards
The estimate of a and m given in the Table 1 explains
the difference in growth pattern of the malformed
and normal panicles. The major difference in a value
of both types of panicles is apparent and indicates
that the final length which can be attained by the
malformed panicle is about 66.67% that of normal
panicle.
The present data is one of the classical example
where Gompertz function has inadequately fitted
normal panicle data during initial growth period (up
to 15 days). This underestimated the growth rate
during the period. The.estimate of final length of the
panicle is more in case of Gompertz and logistic
function as compared to Richards function (Table 2).
Logistic and Gompertz equations both overestimated
the growth rate of malformed panicle during 35th to
50th day and resulted higher final growth estimates.
Despite Richards function was introduced in 1959,
came in extended use considerably late with the
increased use of personal computers in numerical
analysis. Causton et al. (1978) have shown that it is
a reasonable model for the determinate nature of
growth. It has been used to elucidate pattern of leaf
growth (Dennett et al., 1978) which has growth curve
shape similar to mango panicles. Richards model is
based on biologically realistic model and has many
advantageous uses for growth analysis of mango
panicles at different time and treatments.
The better results obtained with the use of Richards
model is due to its flexibility for fitting of data of
diverse origin. It can be reduced to logistic and

3. 64 Journal of Applied Horticulture
Gompertz models also with the change in m. The
logistic and Gompertz equations are also the special
case of Richards function (Richards, 1969).
The functional analysis of data indicated that
Richards function is a suitable model for
summarising panicle growth. The model was found
to be superior over logistic and Gompertz because of
its greater efficiency. Richards function can be
successfully used for simulating panicle growth
under different conditions and treatments.
References
Causton, D.R., C.O. Elias and P. Hadley, 1978. Biometrical
studies of plant growth. I The Richards Function and
its application in analysing the effect of temperature on
leaf growth. Pi. Cell. Enuiron., 1:163-184.
Dennett, M.D., B.AAuld and J. Elston, 1978. A description
of leaf growth in Viciafaba L. Annals Bot., 42:223-232.
Draper, N. and H Smith, 1981. Applied Regression Analy-
sis. Second Edition, New York, Wiley.
Lalancette,N., L.V. Madden, and M.A.Ellis, 1988. A quan-
titative model for describing the sporulation of Plasmo-
para viticola on grape leaves. Phytopathology,
78(10): 1316-1321).
Marquardt, D.W. 1963. An algorithm for least square
estimation of nonlinear parameters. Jour. Soc. Indust.
Appl. Math., 2:431-441.
Moore, F.D., P.A. Jolliffe, P.C. Stanwood and E.E. Roos,
1988. Use of Richards Function to interpret single seed
conductivity. HortSci, 23(2):396-398.
Pienaar, L.V.·and K.J. Turnbull, 1973. The Chapman-
Richards generalization of Von Bertalanffy's growth
model for basal area growth and yield in even-aged
stands. For. Sci. 19(1):2-22.
Rajan, S. 1987. Biochemical basis of mango (Mangifera
indica L.).malformation. Ph.D. thesis submitted to In-
dian Agricultural Research Institute, New Delhi.
Rajan, Sand P.K. Majumder, 1995. Growth analysis of
malformed mango panicles by using asymptotic curves.
Jour. Rec. Adu. Appl. Sci., 10(1-2):85-86
Richards, F.J. 1969. The quantitative analysis of growth.
PI. Physiol, Vol VA(Ed Steward, F.C.), pp. 3-76.
Sukhvibul, N., A.W. Whiley, M.K. Smith, Suzan E. Heth-
erington and V. Vithanage, 1999. Effect of temperature
on inflorescerlce and floral development in four mango
(Mangifera indica L.) cultivars. Scientia Hortic., 82:67-
84.